[FOM] Re: Borel categoricity

Dave Marker marker at math.uic.edu
Wed Jul 16 15:35:47 EDT 2003

Having not thought about the problem for many years, I realize
that I had forgoten one of the key steps

On Thu, 10 Jul 2003, Dave Marker wrote:
> iv) If T has a two cardinal model then T has a Borel two cardinal model.
> (if you've done i) and ii) it might be helpful to assume omega-stability
> here)

This is known to be true, at least if T is omega-stable (or even stable).
Under these assumptions, if there is a two cardinal model, there are
arbitrarily large two-cardinal models (this is a theorem of Lachlan) and
you can then use Erdos-Rado to build two cardinal models generated by
indiscernibles. Using a set of indiscernibles of order type the reals
yields a Borel two cardinal model.

Thus, to answer the question, is every Borel categorical theory
categorical in every uncountable power, it is enough to answer
the following question:

If T is a complete theory in a countable language, is there a
Borel model M of T such that M realizes only countably many types
over every countable subset.

The argument I sketched above is given in Charles Steinhorn's paper
"Borel Structures and Measure and Category Logics" in
"Model Theoretic Logics" J. Barwise and S. Feferman ed, Springer 1985.

Dave Marker

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